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The motivic Hall product for a one-vertex zero-arrow quiver

Example

Let $Q$ have one vertex and no arrows.

\begin{tikzpicture}
\node[circle,draw,inner sep=1.5pt,label=below:{1}] (v) at (0,0) {};
\end{tikzpicture}

The vertex label is $1$ ; the absence of edges means a representation is only a finite-dimensional vector space. Let $S$ be its one-dimensional representation and let $δ_{S} = [B {GL}_{1} \to M_{1} (Q)]$ be the identity stack function. For every field $k$ , $I_{Q} (δ_{S} * δ_{S}) = L^{- 1} [B {GL}_{1}]^{2} x^{2} .$ On the finite-field groupoid-count specialization $k = F_{q}$ , this coefficient is $q^{- 1} (q - 1)^{- 2}$ .

Facts & Assumptions

Given: The one-vertex zero-arrow quiver, its representation stacks, and the quotient-first motivic Hall product.

[F1]

The Hall product is the stack of sequences $0 \to B \to E \to A \to 0$ with quotient first and subrepresentation second (The motivic Hall algebra of a finite quiver).

[F2]

Its extension-stack relative class is $L^{- χ_{Q} (d, e)}$ (The motive of a quiver extension stack).

[F3]

The integration map sends a stack function $[X \to M_{d}]$ to $[X] x^{d}$ and the target twist is the stated Euler factor (Quiver stack integration is multiplicative with the Euler twist).

[F4]

For a finite-field quotient stack in this construction, groupoid cardinality is the sum of inverse automorphism orders and respects the motivic stack relations (Finite-field groupoid cardinality for quiver stack specializations, Finite-field groupoid cardinality respects the motivic stack relations).

[A1]

The algebraic-stack convention inherits its represented-sheaf descent dependency on AC (Algebraic stacks and their inertia stacks, The Axiom of Choice); the calculation itself uses only finite-dimensional vector spaces and finite groupoids after specialization.

Verification

technique · direct

1.1F1F2F3A1

For this quiver, ${Rep}_{1} (Q)$ is a point and $G_{1} = {GL}_{1}$ , so $M_{1} (Q) = B {GL}_{1}$ . For the two degree-one factors, $d = e = 1$ and $χ_{Q} (1, 1) = 1$ . By [F2], the extension stack over the pair of endpoint stacks has relative class $L^{- 1}$ . Pushing to the middle and then integrating therefore gives $I_{Q} (δ_{S} * δ_{S}) = L^{- 1} [B {GL}_{1}]^{2} x^{2}$ .

2.1F1F2F4algebra∎

Over $F_{q}$ , the groupoid $B {GL}_{1} (F_{q})$ has one isomorphism class with automorphism group of order $q - 1$ , hence groupoid cardinality $(q - 1)^{- 1}$ . The extension fiber over the two fixed endpoints is $B G_{a} (F_{q})$ : the middle sequence splits, and its endpoint-fixing automorphisms are the linear maps $S \to S$ , a group of order $q$ . Its groupoid cardinality is $q^{- 1}$ by [F4]. Multiplying these three factors gives $q^{- 1} (q - 1)^{- 2}$ , as stated.

Remarks

Here $χ_{Q} (1, 1) = 1$ because there is one vertex and no arrows. This is the untwisted Hall stack coefficient with its automorphism group retained; it is not the finite subobject count $q + 1$ used by the separate untwisted Ringel product.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Tom Bridgeland, An Introduction to Motivic Hall Algebras, arXiv:1002.4372v1, §§4.1–4.2 and 5.2 (standard reference, not scraped)
Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2, §§1.1–1.3 (standard reference, not scraped)